"Why would anyone count the beats any different to the one counting on the video."

Because as phil discovered, his count is closer to literally matching the way foxtrot is actually danced than the one you have in mind is.

Of course we have to be careful when we speak of what the difference is. From Phil's commment, what he was actually counting was all of the half beats, which is something that you've often recommended yourself, serendipity. The difference is where he was trying to be at the various points in that count - his targets were informed by a more sophisticated understanding of how foxtrot is actually danced.

We mustn't get our Waltz counting muddled with Foxtrot. Waltz is the one that has the three beats divided into six. To follow John Wood on one section only in the Foxtrot. Once understood it is repeated all the way through. At the end of the third step of the Feather the LF comes along side the RF on an and count. It is sometimes refered to as a balance point. There is another at the end of the third step of the Reverse. And another at the end of the sixth step. All given an and count and are points of balance as well as being were CBM or CBMP comes in. There is another at the end of the quick in the Three Step. And so on and so forth. The move is continuous. So don't think it is stop start. But it is there and can be seen on any tape. You have probably seen what appears to be a dragging of the rear feet on some steps, like a delaying action. The Wave is one place where it shows up.

"We mustn't get our Waltz counting muddled with Foxtrot. Waltz is the one that has the three beats divided into six."

Subdivision is simply a tool, it's not unique to waltz or rumba or foxtrot.

You know, I'm watching your Hilton foxtrot basics video, and if I just watch and listen at the start of the timing/counts take it sure seems as if he's landing those quicks when he says quick. But when I stop it and actually measure the time between the foot placements by counting the frames of video, it's obvious that this is not actually the case, in fact in that particular case the steps are quite close to being equal in duration, which means that only one of the three can be on a beat.

As I said, human perception is notoriously unreliable for determining when things actually happened. But sequence of equally spaced images on a movie film or video does not have that handicap. It's really too bad we can't sit down together, unwind a movie reel across a light board and count the number of frames in each step - when the evidence is under your nose, it really becomes undeniable.

The frame counts for the feather in that take:

513: Right foot begins moving towards slow
534: left foot towards the quick
552: right foot towards the second quick
575: second quick lands

The spacings:

slow: 21 frames
quick: 18 frames
quick: 23 frames

vs a beat lasting about 15-16 frames.

I was talking about a reverse wave, and the list was just that - a list of figures. Not meant to be a sequence of figures.
It could be argued that the intervals between 1&, 3, 4& are the same as those between 1, 2&, 4. i.e. take the 'standard' 1,3,4 and land the second step half a beat early.
I tried that at first, because it's easier, but 1&, 3, 4& matches the music better. However, i'm still learning.
I'll repeat what i said earlier, the Hilton Basic foxtrot is just that - a BASIC foxtrot.
Have a look at this Andrew Sinkinson clip:
http://www.youtube.com/watch?v=c0K838NZvY0&feature=related

Phil. I haven't yet looked at the Andrew clip. But I can't see how I could possibly manage to have the presence of mind to consistently land any step half a beat early. If I did I am deliberately trying to step inbetween beats. With music playing 1 2 3 4 it seems to me I would have trouble placing any step half a beat closer to the next step, or half a beat away from the rear step, depending if I look at it as .Is the cup half full or half empty.
What seems to be confusing to some is the (and) after the third step of the Feather is not a step. It is a position of the body coming into a position of balance.What I have been taught is get those quick steps on the beat . Make it smooth. The slows will look after themselves. Do you see any demomnstration in a Reverse or Natural Weave where the steps are not right on the beat.The other thing that does need to be looked at is CBMP on the third step. What some people think is CBMP is not because they have turned their body slightly anti -clock without realising it. If we go to another dance the Progressive Chasse in the Quickstep steps 2 3 4 keep the body practically square to the wall the whole time otherwise the bodies will become out of alignment on the outside step. Unfortunately most as in the Foxtrot they turn their bodies to the left destroying what should be CBMP.
I believe the advise spoken by John Wood for the Foxtrot which is. Stay on the suporting leg longer. Combine that with get the quicks right on the beat is all that is needed. Good Luck.

"What I have been taught is get those quick steps on the beat ."

Then you have been taught to dance foxtrot differently from everyone else.

That's simply not the way it's done - even on the videos that you try to cite as an example of this, measurement shows far too much time between the quicks for them both to be landing on beats.

"Do you see any demomnstration in a Reverse or Natural Weave where the steps are not right on the beat."

Most certainly. The sequence of repeated quicks in the weave action itself may fall nearly on beats, but the final QQ of the weave ending will have a similar timing to the quicks of a SQQ group, which is to say the penultimate quick landing near beat three and the final one landing long after beat four.

If you want to dance in between the quicks go ahead that's your choice. It remains bad timing. I still don't see the logic here. If I move one step imbetween the beat do I also move the other step in between it and the next step also. Or do I place one of those steps almost on the one in front. I may be wrong but if I step in between a beat and don't step in between the next beat. If I step the second one on the beat they will only be 1/4 of a beat apart. What happens next. Tell me who on any tape or in any manuel gives instructions that says Don't step on the beat as it is being played. Why not turn the music off and dance with nothing to guide you at all..

"If you want to dance in between the quicks go ahead that's your choice. It remains bad timing."

Call it whatever you want, but it's easy to determine that someone who does this will be in very good company, as all of the worlds best dancers drift their final quicks beyond the beat. They'd look very silly if they didn't.

"If I move one step imbetween the beat do I also move the other step in between it and the next step also."

You spread the three steps out more equally over the four beats. None of the intervals are as small as a beat, they are all larger. Very often the first quick lands on beat three, but that's the only step in SQQ group that actually lands on a beat.

"Tell me who on any tape or in any manuel gives instructions that says Don't step on the beat as it is being played."

It's not a topic that usually gets a lot of literal comment, but if you find a knowledgeable teacher willing to answer the question literally (and accurately) they will tell you something comparable to what can be measured on any tape of good foxtrot, which is to give the final quick more than a beat worth of time so that it lands well after beat four rather than on it. This mans that the slow gets somewhat less than two beats, and also does not land on a beat.

Most teachers don't like to talk about it that way though; they'd rather that you learn the feel of drifted foxtrot rather than the math that precisely describes it.

"Why not turn the music off and dance with nothing to guide you at all.."

Because this in-between-the-beats foot timing is what enables you to dance foxtrot with your body in time with the music. Foxtrot is not a dance that expresses the beats in the feet, it's a dance that expresses the measures in the body swing. It's not about the feet!

A Chinese friend of mine asked if I would like to see this video he had. On it the couple looked imaculate in poise and their movement was excellent. But it was a Foxtrot and that danced completely out of time. It was an absolute mess. Then it dawned on me that nobody could look that good and be so far out with their timing. What had happend I think the music and the video had been put together afterwards and the person who was supposed to co-ordinate the two had n't a clue. If you ever watched Singing in the Rain you would know what the end result can be.
How would you combine the two if it was your job to put it together.
I would set the music to any of the quicks of which I know in the Feather there are two. I most certainly would not set it to miss the beats. Is that what you would do.

Just as an aside. Because of the relatively slow speed of sound, in a large venue the dancers may hear the music around 1/10 of second after it is recorded on the video of them dancing. So even if they did land the quick when they heard the beat it would look late.